This article deals with a new type of convex reformulations of quadratically constrained quadratic optimization problems (QCQPs). General QCQPs are hard, and form a large and quite versatile class of optimization problems. Neither the objective function nor the feasible set of a QCQP need to be convex, the latter may even be disconnected. One way to (approximately) solve QCQPs is to look for convex relaxations which in the best case yield exact reformulations. A well established framework of achieving such reformulation is based on the set \({\mathcal {G}}({\mathcal {F}}) \mathrm {clconv} \left\lbrace \left(\mathbf {x},\mathbf {x}\mathbf {x})\colon \mathbf {x}\in {\mathcal {F}} \right\rbrace \right.\)  where \({\mathcal {F}}\)  is the feasible set of a QCQP and \(\mathrm {clconv}(\cdot )\)  is the is the closure of the convex hull of a set. For examples of such reformulations see [1]}, [2]}, [3]}, [4]}, [5]}, [6]}, [7]}, where [6]} contains a detailed description of the use of \({\mathcal {G}}({\mathcal {F}})\) . These reformulations typically involve set-completely positive matrix cones given by
\(\mathcal {CPP}_n({\mathcal {K}}) &\left\lbrace \sum _{i=1}^{k}\mathbf {x}_i\mathbf {x}_i\mathbf {x}_i \in {\mathcal {K}},\ i\in [1\!:\!k]\right\rbrace \\ &= \mathrm {clconv}\left\lbrace \mathbf {x}\mathbf {x}\mathbf {x}\in {\mathcal {K}}\right\rbrace = \lbrace {\mathsf {X}}{\mathsf {X}}{\mathsf {X}}\in \mathbb {R}^{n\times k}\, , \, \mathbf {x}_i \in {\mathcal {K}}, \, i \in [1\!:\!k] \rbrace ,\) 
